with_o my_o compass_n i_o take_v the_o length_n of_o a._n b._n and_o set_v the_o one_o foot_n of_o my_o compass_n in_o c_o and_o draw_v a_o arch_a line_n with_o the_o other_o foot_n likewaies_n i_o take_v the_o length_n of_o e._n f_o and_o set_v one_o foot_n in_o d_o and_o with_o the_o other_o foot_n i_o make_v a_o arch_a line_n cross_v the_o other_o arch_n and_o the_o prick_n of_o their_o meeting_n which_o be_v g._n shall_v be_v the_o third_o corner_n of_o the_o triangle_n for_o in_o all_o such_o kynde_n of_o work_v to_o make_v a_o tryangle_n if_o you_o have_v one_o line_n draw_v there_o remain_v nothing_o else_o but_o to_o find_v where_o the_o pitch_n of_o the_o third_o corner_n shall_v be_v for_o two_o of_o they_o must_v needs_o be_v at_o the_o two_o cande_n of_o the_o line_n that_o be_v draw_v the_o xiii_o conclusion_n if_o you_o have_v a_o line_n appoint_v and_o a_o point_n in_o it_o limit_v how_o you_o may_v make_v on_o it_o a_o right_a line_v angle_n equal_a to_o a_o other_o right_o line_v angle_n all_o ready_a assign_v first_o draw_v a_o line_n against_o the_o corner_n assign_v and_o so_o be_v it_o a_o triangle_n then_o take_v heed_n to_o the_o line_n and_o the_o point_n in_o it_o assign_v and_o consider_v if_o that_o line_n from_o the_o prick_n to_o this_o end_n be_v as_o long_o as_o any_o of_o the_o side_n that_o make_v the_o triangle_n assign_v and_o if_o it_o be_v long_o enough_o then_o prick_v out_o there_o the_o length_n of_o one_o of_o the_o line_n and_o then_o work_v with_o the_o other_o two_o line_n accord_v to_o the_o last_o conclusion_n make_v a_o triangle_n of_o three_o like_a line_n to_o that_o assign_a triangle_n if_o it_o be_v not_o long_o enough_o thenne_v lengthen_v it_o first_o and_o afterward_o do_v as_o i_o have_v say_v before_o example_n let_v the_o angle_n appoint_v be_v a._n b._n c_o and_o the_o corner_n assign_v b._n farthermore_o let_v the_o limit_v line_n be_v d._n g_o and_o the_o prick_v assign_v d._n first_o therefore_o by_o draw_v the_o line_n a._n c_o i_o make_v the_o triangle_n a.b.c._n then_o consider_v that_o d._n g_o be_v long_a thanne_n a._n b_o you_o shall_v cut_v out_o a_o line_n from_o d._n towards_o g_o equal_a to_o a._n b_o as_z for_o example_n d_o f._n then_o measure_v out_o the_o other_o ij_o line_n and_o work_n with_o they_o according_a as_o the_o conclusion_n with_o the_o first_o also_o and_o the_o second_o teach_v you_o and_o then_o have_v you_o do_v the_o xiiii_o conclusion_n to_o make_v a_o square_a quadrate_n of_o any_o right_a line_n appoint_v first_o make_v a_o plumbe_v line_n unto_o your_o line_n appoint_v which_o shall_v light_v at_o one_o of_o the_o end_n of_o it_o accord_v to_o the_o five_o conclusion_n and_o let_v it_o be_v of_o like_a length_n as_o your_o first_o line_n be_v then_o open_a your_o compass_n to_o the_o just_a length_n of_o one_o of_o they_o and_o set_v one_o foot_n of_o the_o compass_n in_o the_o end_n of_o the_o one_o line_n and_o with_o the_o other_o foot_n draw_v a_o arch_n line_n there_o as_o you_o think_v that_o the_o fowerth_o corner_n shall_v be_v after_o that_o set_v the_o one_o foot_n of_o the_o same_o compass_n unstur_v in_o the_o cande_v of_o the_o other_o line_n and_o draw_v a_o other_o arch_n line_n across_o the_o first_o arch_n line_n and_o the_o point_n that_o they_o do_v cross_v in_o be_v the_o prick_n of_o the_o four_o corner_n of_o the_o square_a quadrate_n which_o you_o seek_v for_o therefore_o draw_v a_o line_n from_o that_o prick_n to_o the_o eande_n of_o each_o line_n and_o you_o shall_v thereby_o have_v make_v a_o square_a quadrate_n example_n a._n b._n be_v the_o line_n propose_v of_o which_o i_o shall_v make_v a_o square_a quadrate_n therefore_o first_o i_o make_v a_o plumbe_v line_n unto_o it_o which_o shall_v light_o in_o a_o and_o that_o plumb_n line_n be_v a._n b_o then_o open_v i_o my_o compass_n as_o wide_o as_o the_o length_n of_o a._n b_o or_o b._n c_o for_o they_o must_v be_v both_o equal_a and_o i_o set_v the_o one_o foot_n of_o the_o end_n in_o c_o and_o with_o the_o other_o i_o make_v a_o arch_n line_n nigh_o unto_o d_z afterward_o i_o set_v the_o compass_n again_o with_o one_o foot_n in_o b_o a_z and_z with_o the_o other_o foot_n i_o make_v a_o arch_n line_n across_o the_o first_o arch_n line_n in_o d_o and_o from_o the_o prick_n of_o their_o cross_n i_o draw_v two_o line_n one_o to_o b_o a_o and_o a_o other_o to_o c_o and_o so_o have_v i_o make_v the_o square_a quadrate_n that_o i_o intend_v the_o xv._o conclusion_n to_o make_v a_o likeiamme_n equal_a to_o a_o triangle_n appoint_v and_o that_o in_o a_o right_n line_v angle_n limit_v first_o from_o one_o of_o the_o angle_n of_o the_o triangle_n you_o shall_v draw_v a_o gemowe_a line_n which_o shall_v be_v a_o parallel_n to_o that_o side_n of_o the_o triangle_n on_o which_o you_o will_v make_v that_o likeiamme_n then_o on_o one_o end_n of_o the_o side_n of_o the_o triangle_n which_o lie_v against_o the_o gemowe_a line_n you_o shall_v draw_v forth_o a_o line_n unto_o the_o gemow_n line_n so_o that_o one_o angle_n that_o come_v of_o those_o two_o line_n be_v like_a to_o the_o angle_n which_o be_v limit_v unto_o you_o then_o shall_v you_o divide_v into_o ij_o equal_a part_n that_o side_n of_o the_o triangle_n which_o bear_v that_o line_n and_o from_o the_o prick_n of_o that_o division_n you_o shall_v raise_v a_o other_o line_n parallel_n to_o that_o former_a line_n and_o continewe_v it_o unto_o the_o first_o gemowe_n line_n and_o they_o of_o those_o two_o last_o gemowe_n line_n and_o the_o first_o gemowe_n line_n with_o the_o half_a side_n of_o the_o triangle_n be_v make_v a_o lykeiamme_n equal_a to_o the_o triangle_n appoint_v and_o have_v a_o angle_n like_o to_o a_o angle_n limit_v accord_v to_o the_o conclusion_n example_n b._n c._n g_o be_v the_o triangle_n appoint_v unto_o which_o i_o must_v make_v a_o equal_a likeiamme_n and_o d_o be_v the_o angle_n that_o the_o likeiamme_n must_v have_v therefore_o first_o intend_v to_o erect_v the_o likeiamme_n on_o the_o one_o side_n that_o the_o ground_n line_n of_o the_o triangle_n which_o be_v b._n g._n i_o do_v draw_v a_o gemow_n line_n by_o c_o and_o make_v it_o parallel_v to_o the_o ground_n line_n b._n g_o and_o that_o new_a gemow_n line_n be_v a._n h._n then_o do_v i_o raise_v a_o line_n from_o b._n unto_o the_o gemowe_a line_n which_o line_n be_v a._n b_o and_o make_v a_o angle_n equal_a to_o d_o that_o be_v the_o appoint_a angle_n accord_v as_o the_o eight_o conclusion_n teach_v and_o that_o angle_n be_v b._n a._n e._n then_o to_o proceed_v i_o do_v part_n in_o the_o middle_n the_o say_a ground_n line_n b._n g_o in_o the_o prick_n f_o from_o which_o prick_n i_o draw_v to_o the_o first_o gemowe_n line_n a._n h._n a_o other_o line_n that_o be_v parallale_n to_o a._n b_o and_o that_o line_n be_v e._n f._n now_o say_v i_o that_o the_o likeiamme_n b._n a._n e._n f_o be_v equal_a to_o the_o triangle_n b._n c._n g._n and_o also_o that_o it_o have_v one_o angle_n that_o be_v b._n a._n e._n like_v to_o d._n the_o angle_n that_o be_v limit_v and_o so_o have_v i_o my_o intent_n the_o prose_n of_o the_o equalnes_n of_o those_o two_o figure_n do_v depend_v of_o the_o xli_o proposition_n of_o euclides_n first_o book_n and_o be_v the_o xxxi_o proposition_n of_o this_o second_o book_n of_o theoremis_n which_o say_v that_o when_o a_o tryangle_n and_o a_o likeiamme_n be_v make_v between_o two_o self_n same_o gemow_n line_n and_o have_v their_o ground_n line_n of_o one_o length_n then_o be_v the_o likeiamme_n double_a to_o the_o triangle_n whereof_o it_o follow_v that_o if_o two_o such_o figure_n so_o draw_v differ_v in_o their_o ground_n line_n only_o so_o that_o the_o ground_n line_n of_o the_o likeiamme_n be_v but_o half_a the_o ground_n line_n of_o the_o triangle_n then_o be_v those_o two_o figure_n equal_a as_o you_o shall_v more_o at_o large_a perceive_v by_o the_o book_n of_o theoremis_n in_o the_o xxxi_o theorem_a the_o xvi_n conclusion_n to_o make_v a_o likeiamme_n equal_a to_o a_o triangle_n appoint_v accord_v to_o a_o angle_n limit_v and_o on_o a_o line_n also_o assign_v in_o the_o last_o conclusion_n the_o side_n of_o your_o likeiamme_n be_v leave_v to_o your_o liberty_n though_o you_o have_v a_o angle_n appoint_v now_o in_o this_o conclusion_n you_o be_v somewhat_o more_o restrain_v of_o liberty_n since_o the_o line_n be_v limit_v which_o must_v be_v the_o side_n of_o the_o likeiamme_n therefore_o thus_o shall_v you_o proceed_v first_o accord_v to_o the_o last_o conclusion_n make_v a_o likeiamme_n
figure_n with_o a_o right_a angle_n accord_v to_o the_o xu_o conclusion_n then_o consider_v the_o likeiamme_n whether_o it_o have_v all_o his_o side_n equal_a or_o not_o for_o if_o they_o be_v all_o equal_a then_o have_v you_o do_v your_o conclusion_n but_o and_o if_o the_o side_n be_v not_o all_o equal_a then_o shall_v you_o make_v one_o right_a line_n just_a as_o long_o as_o two_o of_o those_o unequal_a side_n that_o line_n shall_v you_o divide_v in_o the_o middle_n and_o on_o that_o prick_n draw_v half_a a_o circle_n then_o cut_v from_o that_o diameter_n of_o the_o half_a circle_n a_o certain_a portion_n equal_a to_o the_o one_o side_n of_o the_o likeiamme_n and_o from_o that_o point_n of_o division_n shall_v you_o erect_v a_o perpendicular_a which_o shall_v touch_v the_o edge_n of_o the_o circle_n and_o that_o perpendicular_a shall_v be_v the_o just_a side_n of_o the_o square_a quadrate_n equal_a both_o to_o the_o lykeiamme_n and_o also_o to_o the_o right_n line_v figure_n appoint_v as_o the_o conclusion_n will_v example_n k_o be_v the_o right_n line_v figure_n appoint_v and_o b._n c._n d._n e_o be_v the_o likeiamme_n with_o right_a angle_n equal_a unto_o k_o but_o because_o that_o this_o likeiamme_n be_v not_o a_o square_a quadrate_n i_o must_v turn_v it_o into_o such_o one_o after_o this_o sort_n i_o shall_v make_v one_o right_a line_n as_o long_o as_o two_o unequal_a side_n of_o the_o likeiamme_n that_o line_n here_o be_v f._n g_o which_o be_v equal_a to_o b._n c_o and_o c._n e._n then_o part_n i_o that_o line_n in_o the_o middle_n in_o the_o prick_n m_o and_o on_o that_o prick_n i_o make_v half_o a_o circle_n accord_v to_o the_o length_n of_o the_o diameter_n f._n g._n afterward_o i_o cut_v away_o a_o piece_n from_o f._n g_o equal_a to_o c._n e_o mark_n that_o point_n with_o h._n and_o on_o that_o prick_n i_o erect_v a_o perpendicular_a h._n k_o which_o be_v the_o just_a side_n to_o the_o square_a quadrate_n that_o i_o seek_v for_o therefore_o accord_v to_o the_o doctrine_n of_o the_o ten_o conclusion_n of_o that_o line_n i_o do_v make_v a_o square_a quadrate_n and_o so_o have_v i_o attain_v the_o practice_n of_o this_o conclusion_n the_o xx._n conclusion_n when_o any_o two_o square_a quadrate_n be_v set_v forth_o how_o you_o may_v make_v one_o equal_a to_o they_o both_o first_o draw_v a_o right_a line_n equal_a to_o the_o side_n of_o one_o of_o the_o quadrate_n and_o on_o the_o end_n of_o it_o make_v a_o perpendicular_a equal_a in_o length_n to_o the_o side_n of_o the_o other_o quadrate_n then_o draw_v a_o bias_n line_n between_o those_o two_o other_o line_n make_v thereof_o a_o right_n angel_v triangle_n and_o that_o bias_n line_n will_v make_v a_o square_a quadrate_n equal_a to_o the_o other_o two_o quadrate_n appoint_v example_n a.b._n and_o c._n d_o be_v the_o two_o square_a quadrate_n appoint_v unto_o which_o i_o must_v make_v one_o equal_a square_a quadrate_n first_o therefore_o i_o do_v make_v a_o right_a line_n e._n f_o equal_a to_o one_o of_o the_o side_n of_o the_o square_a quadrate_n a.b._n and_o on_o the_o one_o end_n of_o it_o i_o make_v a_o plumbe_v line_n e._n g_o equal_a to_o the_o side_n of_o the_o other_o quadrate_n d._n c._n then_o draw_v i_o a_o bias_n line_n g._n f_o which_o be_v make_v the_o side_n of_o a_o quadrate_n accord_v to_o the_o ten_o conclusion_n will_v accomplish_v the_o work_n of_o this_o practice_n for_o the_o quadrate_n h._n be_v as_o much_o just_a as_o the_o other_o two_o i_o mean_v a._n b._n and_o d._n c._n the_o xxi_o conclusion_n when_o any_o two_o quadrate_n be_v set_v forth_o how_o to_o make_v a_o squire_n about_o the_o one_o quadrate_n which_o shall_v be_v equal_a to_o the_o other_o quadrate_n determine_v with_o yourself_o about_o which_o quadrate_n you_o will_v make_v the_o squire_n and_o draw_v one_o side_n of_o that_o quadrate_n forth_o in_o lengte_n accord_v to_o the_o measure_n of_o the_o side_n of_o the_o other_o quadrate_n which_o line_n you_o may_v call_v the_o ground_n line_n and_o then_o have_v you_o a_o right_a angle_n make_v on_o this_o line_n by_o a_o other_o side_n of_o the_o same_o quadrate_n therefore_o turn_v that_o into_o a_o right_a corner_a triangle_n accord_v to_o the_o work_n in_o the_o last_o conclusion_n by_o make_v of_o a_o bias_n line_n and_o that_o bias_n line_n will_v perform_v the_o work_n of_o your_o desire_n for_o if_o you_o take_v the_o length_n of_o that_o bias_n line_n with_o your_o compass_n and_o then_o set_v one_o foot_n of_o the_o compass_n in_o the_o far_a angle_n of_o the_o first_o quadrate_n which_o be_v the_o one_o end_n of_o the_o groundline_a and_o extend_v the_o other_o foot_n on_o the_o same_o line_n accord_v to_o the_o measure_n of_o the_o bias_n line_n and_o of_o that_o line_n make_v a_o quadrate_n enclose_n the_o first_o quadrate_n then_o will_v there_o appear_v the_o form_n of_o a_o squire_n about_o the_o first_o quadrate_n which_o squire_n be_v equal_a to_o the_o second_o quadrate_n example_n the_o first_o square_a quadrate_n be_v a._n b._n c._n d_o and_o the_o second_o be_v e._n now_o will_v i_o make_v squire_n about_o the_o quadrate_n a._n b._n c._n d_o which_o shall_v be_v equal_a unto_o the_o quadrate_n e._n therefore_o first_o i_o draw_v the_o line_n a._n d_o more_o in_o length_n accord_v to_o the_o measure_n of_o the_o side_n of_o e_o as_o you_o see_v from_o d._n unto_o f_o and_o so_o the_o hole_n line_n of_o both_o these_o several_a side_n be_v a._n f_o they_o make_v i_o a_o bias_n line_n from_o c_o to_o f_o which_o bias_n line_n be_v the_o measure_n of_o this_o work_n wherefore_o i_o open_v my_o compass_n accord_v to_o the_o length_n of_o that_o bias_n line_n c._n f_o and_o set_v the_o one_o compass_n foot_n in_o a_o and_o extend_v other_o foot_v of_o the_o compass_n towards_o f_o make_v this_o prick_n g_o from_o which_o i_o erect_v a_o plumbe_v line_n g._n h_n and_o so_o make_v out_o the_o square_a quadrate_n a._n g._n h._n k_o who_o side_n be_v equal_a each_o of_o they_o to_o a._n g._n and_o this_o square_n do_v contain_v the_o first_o quadrate_n a._n b._n c._n d_o and_o also_o a_o squire_n g._n h.k_n which_o be_v equal_a to_o the_o second_o quadrate_n e_o for_z as_z the_o last_o conclusion_n declare_v the_o quadrate_n a._n g.h._n k_o be_v equal_a to_o both_o the_o other_o quadrate_v propose_v that_o be_v a._n b._n c.d_n and_o e._n then_o must_v the_o squire_n g._n h.k_n needs_o be_v equal_a to_o e_o consider_v that_o all_o the_o rest_n of_o that_o great_a quadrate_n be_v nothing_o else_o but_o the_o quadrate_n self_n a._n b._n c._n d_o and_o so_o have_v i_o thintent_v of_o this_o conclusion_n the_o xxi_o conclusion_n to_o find_v out_o the_o centre_n of_o any_o circle_n assign_v draw_v a_o cord_n or_o string_n line_n across_o the_o circle_n then_o divide_v into_o two_o equal_a part_n both_o that_o cord_n and_o also_o the_o bow_n line_n or_o arch_n line_n that_o serve_v to_o that_o cord_n and_o from_o the_o prick_v of_o those_o division_n if_o you_o draw_v a_o other_o line_n across_o the_o circle_n it_o must_v needs_o pass_v by_o the_o centre_n therefore_o divide_v that_o line_n in_o the_o middle_n and_o that_o middle_a prick_n be_v the_o centre_n of_o the_o circle_n propose_v example_n let_v the_o circle_n be_v a._n b.c.d_n who_o centre_n i_o shall_v seek_v first_o therefore_o i_o draw_v a_o cord_n across_o the_o circle_n that_o be_v a._n c._n then_o do_v i_o divide_v that_o cord_n in_o the_o middle_n in_o e_o and_o likewaies_o also_o do_v i_o divide_v his_o arch_n line_n a._n b.c_n in_o the_o middle_n in_o the_o point_n b._n afterward_o i_o draw_v a_o line_n from_o b._n to_o e_z and_o so_o cross_v the_o circle_n which_o line_n be_v b._n d_o in_o which_o line_n be_v the_o centre_n that_o i_o seek_v for_o therefore_o if_o i_o part_n that_o line_n b._n d_o in_o the_o middle_n in_o to_o two_o equal_a portion_n that_o middle_a prick_n which_o here_o be_v f_o be_v the_o very_a centre_n of_o the_o say_a circle_n that_o i_o seek_v this_o conclusion_n may_v other_o way_n be_v wrought_v as_o the_o most_o part_n of_o conclusion_n have_v sundry_a form_n of_o practice_n and_o that_o be_v by_o make_v three_o prick_v in_o the_o circumference_n of_o the_o circle_n at_o liberty_n where_o you_o will_v and_o then_o find_v the_o centre_n to_o those_o three_o prick_v which_o work_n because_o it_o serve_v for_o sundry_a use_n i_o think_v meet_v to_o make_v it_o a_o several_a conclusion_n by_o itself_o the_o xxiii_n conclusion_n to_o find_v the_o come_v centre_n belong_v to_o any_o three_o prick_v appoint_v if_o they_o be_v not_o in_o a_o exact_v right_a line_n
the_o angle_n c_o be_v equal_a to_o d._n the_o nine_o theorem_a when_o so_o ever_o in_o any_o triangle_n the_o line_n of_o one_o side_n be_v draw_v forth_o in_o length_n that_o utter_a angle_n be_v great_a than_o any_o of_o the_o two_o inner_a corner_n that_o join_v not_o with_o it_o example_n the_o triangle_n a._n d._n c_o have_v his_o ground_n line_n a._n c._n draw_v forth_o in_o length_n unto_o b_o so_o that_o the_o utter_a corner_n that_o it_o make_v at_o c_o be_v great_a than_o any_o of_o the_o two_o inner_a corner_n that_o lie_v against_o it_o and_o join_v not_o with_o it_o which_o be_v a._n and_o d_o for_o they_o both_o be_v lesser_a than_o a_o right_a angle_n and_o be_v sharp_a angle_n but_o c._n be_v a_o blonte_a angle_n and_o therefore_o great_a than_o a_o right_a angle_n the_o ten_o theorem_a in_o every_o triangle_n any_o two_o corner_n how_o so_o ever_o you_o take_v they_o be_v less_o they_o ij_o right_n corner_n example_n in_o the_o first_o triangle_n e_o which_o be_v a_o threlyke_a and_o therefore_o have_v all_o his_o angle_n sharp_a take_v any_o two_o corner_n that_o you_o will_v and_o you_o shall_v perceive_v that_o they_o be_v lesser_a than_o two_o right_a corner_n for_o in_o every_o triangle_n that_o have_v all_o sharp_a corner_n as_o you_o see_v it_o to_o be_v in_o this_o example_n every_o corner_n be_v less_o than_o a_o right_a corner_n and_o therefore_o also_o every_o two_o corner_n must_v needs_o be_v less_o than_o two_o right_a corner_n furthermore_o in_o that_o other_o triangle_n mark_v with_o m_n which_o have_v two_o sharp_a corner_n and_o one_o right_a any_o two_o of_o they_o also_o be_v less_o than_o two_o right_a angle_n for_o though_o you_o take_v the_o right_a corner_n for_o one_o yet_o the_o other_o which_o be_v a_o sharp_a corner_n be_v less_o than_o a_o right_a corner_n and_o so_o it_o be_v true_a in_o all_o kind_n of_o triangle_n as_o you_o may_v perceive_v more_o plain_o by_o the_o xxij_o theorem_a the_o xi_o theorem_a in_o every_o triangle_n the_o great_a side_n lie_v against_o the_o great_a angle_n example_n as_o in_o this_o triangle_n a._n b._n c_o the_o great_a angle_n be_v c._n and_o a._n b._n which_o be_v the_o side_n that_o lie_v against_o it_o be_v the_o great_a and_o long_a side_n and_o contrary_a way_n as_o a._n c._n be_v the_o short_a side_n so_o b._n which_o be_v the_o angle_n li_v against_o it_o be_v the_o small_a and_o sharp_a angle_n for_o this_o do_v follow_v also_o that_o as_o the_o long_a side_n lie_v against_o the_o great_a angle_n so_o it_o that_o follow_v the_o twelve_o theorem_a in_o every_o triangle_n the_o great_a angle_n lie_v against_o the_o long_a side_n for_o these_o ij_o theorem_n be_v one_o in_o truth_n the_o thirteen_o theorem_a in_o every_o triangle_n any_o ij_o side_n together_o how_o so_o ever_o you_o take_v they_o be_v long_o they_o the_o third_o for_o example_n you_o shall_v take_v this_o triangle_n a.b.c._n which_o have_v a_o veery_a blunt_a corner_n and_o therefore_o one_o of_o his_o side_n great_a a_o good_a deal_n then_o any_o of_o the_o other_o and_o yet_o thr_fw-mi ij_o lesser_a side_n together_o be_v great_a than_o it_o and_o if_o it_o be_v so_o in_o a_o blunt_a angel_v triangle_n it_o must_v needs_o be_v true_a in_o all_o other_o for_o there_o be_v no_o other_o kind_n of_o triangle_n that_o have_v the_o one_o side_n so_o great_a above_o the_o other_o side_n as_o they_o that_o have_v blunt_a corner_n the_o fourtenth_n theorem_a if_o there_o be_v draw_v from_o the_o end_n of_o any_o side_n of_o a_o triangle_n two_o line_n metinge_v within_o the_o triangle_n those_o two_o line_n shall_v be_v less_o than_o the_o other_o two_o side_n of_o the_o triangle_n but_o yet_o the_o corner_n that_o they_o make_v shall_v be_v great_a than_o that_o corner_n of_o the_o triangle_n which_o stand_v over_o it_o example_n a.b.c._n be_v a_o triangle_n on_o who_o ground_n line_n a.b._n there_o be_v draw_v ij_o line_n from_o the_o ij_o end_n of_o it_o i_o say_v from_o a._n and_o b_o and_o they_o meet_v within_o the_o triangle_n in_o the_o point_n d_o wherefore_o i_o say_v that_o as_o those_o two_o line_n a.d._n and_o b._n d_o be_v lesser_a than_o a.c._n and_o b._n c_o so_fw-mi the_o angle_n d._n be_v great_a than_o the_o angle_n c_o which_o be_v the_o angle_n against_o it_o the_o fiftenth_n theorem_a if_o a_o triangle_n have_v two_o side_n equal_a to_o the_o two_o side_n of_o a_o other_o triangle_n but_o yet_o the_o angle_n that_o be_v contain_v between_o those_o side_n great_a than_o the_o like_a angle_n in_o the_o other_o triangle_n then_o be_v his_o ground_n line_v great_a than_o the_o ground_n line_n of_o the_o other_o triangle_n example_n a.b.c._n be_v a_o triangle_n who_o side_n a.c._n and_o b._n c_o be_v equal_a to_o e.d._n and_o d._n f_o the_o two_o side_n of_o the_o triangle_n d._n e._n f_o but_o because_o the_o angle_n in_o d_o be_v great_a than_o the_o angle_n c._n which_o be_v the_o ij_o angle_n contain_v between_o the_o equal_a line_n therefore_o must_v the_o ground_n line_n e._n f._n needs_o be_v great_a thenne_n the_o ground_n line_n a._n b_o as_o you_o see_v plain_o the_o xuj_o theorem_a if_o a_o triangle_n have_v two_o side_n equal_a to_o the_o two_o side_n of_o a_o other_o triangle_n but_o yet_o have_v a_o long_a ground_n line_v they_o that_o other_o triangle_n then_o be_v his_o angle_n that_o lie_v between_o the_o equal_a side_n great_a they_o the_o like_a corner_n in_o the_o other_o triangle_n example_n this_o theorem_a be_v nothing_o else_o but_o the_o sentence_n of_o the_o last_o theorem_a turn_v backward_o and_o therefore_o need_v none_o other_o proof_n neither_o declaration_n than_o the_o other_o example_n the_o sevententh_n theorem_a if_o two_o triangle_n be_v of_o such_o sort_n that_o two_o angle_n of_o the_o one_o be_v equal_a to_o ij_o angle_n of_o the_o other_o and_o that_o one_o side_n of_o the_o one_o be_v equal_a to_o on_o side_n of_o the_o other_o whether_o that_o side_n do_v adjoine_v to_o one_o of_o the_o equal_a corner_n or_o else_o lie_v against_o one_o of_o they_o then_o shall_v the_o other_o two_o side_n of_o those_o triangle_n be_v equal_a together_o and_o the_o third_o corner_n also_o shall_v be_v equal_a in_o those_o two_o triangle_n example_n because_o that_o a._n b._n c_o the_o one_o triangle_n have_v two_o corner_n a._n and_o b_o equal_a to_o d._n e_o that_o be_v two_o corner_n of_o the_o other_o triangle_n d._n e._n f._n and_o that_o they_o have_v one_o side_n in_o theym_n both_o equal_a that_o be_v a._n b_o which_o be_v equaâ—l_o to_o d._n e_o therefore_o shall_v both_o the_o other_o ij_o side_n be_v equal_a one_o to_o a_o other_o as_z a_z c._n and_o b._n c._n equal_a to_o d._n f_o and_o e._n f_o and_o also_o the_o third_o angle_n in_o they_o both_o shall_v be_v equal_v that_o be_v the_o angle_n c._n shall_v be_v equal_a to_o it_o the_o angle_n f._n the_o eightenth_n theorem_a when_o on_o two_o right_a line_n there_o be_v draw_v a_o three_o right_a line_n across_o way_n and_o make_v two_o match_n corner_n of_o the_o one_o line_n equal_a to_o the_o like_o two_o match_n corner_n of_o the_o other_o line_n then_o be_v those_o two_o line_n gemmow_n line_n or_o parallel_n example_n the_o two_o first_o line_n be_v a._n b._n and_o c._n d_o the_o thyrd_o line_n that_o cross_v they_o be_v e._n f._n and_o because_o that_o e._n f._n make_v ij_o match_v angle_n with_o a._n b_o equal_a to_o two_o other_o like_a match_n angle_n on_o c._n d_o that_o be_v to_o say_v e._n g_o equal_a to_o k._n f_o and_o m._n n._n equal_a also_o to_o h_n l._n therefore_o be_v those_o ij_o line_n a._n b._n and_o c._n d._n gemow_n line_n understand_v here_o by_o like_a match_n corner_n those_o that_o go_v one_o way_n as_o do_v e._n g_o and_o k._n f_o lykeways_o n._n m_o and_o h._n l_o for_o as_o e._n g._n and_o h._n l_o other_o n._n m._n and_o k._n f._n go_v not_o one_o way_n so_o be_v not_o they_o like_o match_n corner_n the_o nyntenth_n theorem_a when_o on_o two_o right_a line_n there_o be_v draw_v a_o third_o right_a line_n across_o way_n and_o make_v the_o ij_o over_o corner_n toward_o one_o hand_n equal_a together_o then_o be_v those_o two_o line_n parallel_n and_o in_o like_a manner_n if_o two_o inner_a corner_n towards_o one_o hand_n be_v equal_a to_o ii_o right_a angle_n example_n as_o the_o theorem_a do_v speak_v of_o two_o over_o angle_n so_o must_v you_o understande_v also_o of_o two_o nether_a angle_n for_o the_o judgement_n be_v like_a in_o both_o take_v for_o a_o example_n the_o figure_n of_o the_o last_o theorem_a where_o a._n b_o
and_o c._n d_o be_v call_v parallel_n also_o because_o e._n and_o k_o which_o be_v two_o over_o corner_n be_v equal_a and_o like_o way_n l._n and_o m._n and_o so_o be_v in_o like_a manner_n the_o nether_a corner_n n._n and_o h_n and_o g._n and_o f._n now_o to_o the_o second_o part_n of_o the_o theorem_a those_o two_o line_n a._n b._n and_o c._n d_o shall_v be_v call_v parallel_n because_o the_o ij_o inner_a corner_n as_o for_o example_n those_o two_o that_o be_v towards_o the_o right_a hand_n that_o be_v g._n and_o l._n be_v equal_a by_o the_o first_o part_n of_o this_o nyntenth_n theorem_a therefore_o must_v g._n and_o l._n be_v equal_a to_o two_o right_a angle_n the_o xx_o theorem_a when_o a_o right_a line_n be_v draw_v cross_n over_o two_o right_a gemow_n line_n it_o make_v two_o match_n corner_n of_o the_o one_o line_n equal_a to_o two_o match_n corner_n of_o the_o other_o line_n and_o also_o both_o over_o corner_n of_o one_o hand_n equal_a together_o and_o both_o nether_a corner_n likewaies_o and_o more_o over_o two_o inner_a corner_n and_o two_o utter_a corner_n also_o toward_o one_o hand_n equal_a to_o two_o right_a angle_n example_n because_o a._n b._n and_o c._n d_o in_o the_o last_o figure_n be_v parallel_n therefore_o the_o two_o match_n corner_n of_o the_o one_o line_n as_o e._n g._n be_v equal_a unto_o the_o two_o match_n corner_n of_o the_o other_o line_n that_o be_v k._n f_o and_o lykewaies_o m._n n_n equal_a to_o h._n l._n and_o also_o e._n and_o k._n both_o over_o corner_n of_o the_o left_a hand_n equal_a together_o and_o so_o be_v m._n and_o l_o the_o two_o over_o corner_n on_o the_o right_n hand_n in_o like_a manner_n n._n and_o h_n the_o two_o nether_a corner_n on_o the_o left_a hand_n equal_a each_o to_o other_o and_o g._n and_o f._n the_o two_o nether_a angle_n on_o the_o right_a hand_n equal_v together_o ¶_o farthermore_o yet_o g._n and_o l._n the_o two_o inner_a angle_n on_o the_o right_a hand_n be_v equal_a to_o two_o right_a angle_n and_o so_o be_v m._n and_o f._n the_o two_o utter_a angle_n on_o the_o same_o hand_n in_o like_a manner_n shall_v you_o say_v of_o n._n and_o k._n the_o two_o inner_a corner_n on_o the_o left_a hand_n and_o of_o e._n and_o h._n the_o two_o utter_a corner_n on_o the_o same_o hand_n and_o thus_o you_o see_v the_o agreeable_a sentence_n of_o these_o three_o theorem_n to_o tend_v to_o this_o purpose_n to_o declare_v by_o the_o angle_n how_o to_o judge_v parallel_n and_o contrary_a way_n how_o you_o may_v by_o parallel_n judge_v the_o proportion_n of_o the_o angle_n the_o xxi_o theorem_a what_o so_o ever_o line_n be_v parallel_n to_o any_o other_o line_n those_o same_o be_v parallel_n together_o example_n a._n b._n be_v a_o gemow_n line_n or_o a_o parellele_n unto_o c._n d._n and_o e._n f_o like_o way_n be_v a_o parallel_n unto_o c._n d._n wherefore_o it_o follow_v that_o a._n b._n must_v needs_o be_v a_o parallel_n unto_o e._n f._n the_o xxij_o theorem_a in_o every_o triangle_n when_o any_o side_n be_v draw_v forth_o in_o length_n the_o utter_a angle_n be_v equal_a to_o the_o ij_o inner_a angle_n that_o lie_v against_o it_o and_o all_o iij._o inner_a angle_n of_o any_o triangle_n be_v equal_a to_o ij_o right_n angle_n example_n the_o triangle_n be_v a._n d._n e._n and_o the_o side_n a._n e._n draw_v forth_o unto_o b_z there_o be_v make_v a_o utter_a corner_n which_o be_v c_o and_o this_o utter_a corner_n c_o be_v equal_a to_o both_o the_o inner_a corner_n that_o lie_v against_o it_o which_o be_v a._n and_o d._n and_o all_o three_o inner_a corner_n that_o be_v to_o say_v a._n d._n and_o e_o be_v equal_a to_o two_o right_a corner_n whereof_o it_o follow_v that_o all_o the_o three_o corner_n of_o any_o one_o triangle_n be_v equal_a to_o all_o the_o three_o corner_n of_o everye_o other_o triangle_n for_o what_o so_o ever_o thing_n be_v equal_a to_o anny_n one_o three_o thing_n those_o same_o be_v equal_a togitther_o by_o the_o first_o common_a sentence_n so_o that_o because_o all_o the_o three_o angle_n of_o every_o triangle_n be_v equal_a to_o two_o right_n angle_n and_o all_o right_n angle_v be_v equal_a together_o by_o the_o four_o request_n therefore_o must_v it_o needs_o follow_v that_o all_o the_o three_o corner_n of_o every_o triangle_n account_v they_o together_o be_v equal_a to_o iij._o corner_n of_o any_o other_o triangle_n take_v all_o together_o the_o xxiii_o theorem_a when_o any_o ij_o right_n line_n do_v touch_v and_o couple_v two_o other_o right_a line_n which_o be_v equal_a in_o length_n and_o parallel_n and_o if_o those_o two_o line_n be_v draw_v toward_o one_o hand_n then_o be_v they_o also_o equal_a together_o and_o parallel_n example_n a._n b._n and_o c._n d._n be_v ij_o right_a line_n and_o parallel_n and_o equal_a in_o length_n and_o they_o be_v touch_v and_o join_v together_o by_o ij_o other_o line_n a._n c._n and_o b._n d_o this_o be_v so_o and_o a._n c._n and_o b._n d._n be_v draw_v toward_o one_o side_n that_o be_v to_o say_v both_o toward_o the_o left_a hand_n therefore_o be_v a_o c._n and_o b._n d._n both_o equal_a and_o also_o parallel_v the_o xxiiij_o theorem_a in_o any_o likeiamme_n the_o two_o contrary_a side_n be_v equal_a together_o and_o so_o be_v each_o two_o contrary_a angle_n and_o the_o bias_n line_n that_o be_v draw_v in_o it_o do_v divide_v it_o into_o two_o equal_a portion_n example_n here_o be_v two_o likeiamme_n join_v together_o the_o one_o be_v a_o long_o square_a a._n b._n e_o and_o the_o other_o be_v a_o losengelike_a d._n c._n e._n f._n which_o ij_o likeiamme_n be_v prove_v equal_a together_o because_o they_o have_v one_o ground_n line_n that_o be_v f._n e_o and_o be_v make_v between_o one_o pair_n of_o gemow_n line_n i_o mean_v a._n d._n and_o e._n h._n by_o this_o theorem_a may_v you_o know_v the_o art_n of_o the_o right_a measuringe_v of_o likeiamme_n as_o in_o my_o book_n of_o measure_v i_o will_v more_o plain_o declare_v the_o xxvi_o theorem_a all_o likeiamme_n that_o have_v equal_a ground_n line_n and_o be_v draw_v between_o one_o pair_n of_o parallel_n be_v equal_a together_o example_n first_o you_o must_v mark_v the_o difference_n between_o this_o theorem_a and_o the_o last_o for_o the_o last_o theorem_a presuppose_a to_o the_o diverse_a likeiamme_n one_o ground_n line_n common_a to_o they_o but_o this_o theorem_a do_v presuppose_v a_o diverse_a ground_n line_n for_o every_o like_a iamme_n only_o mean_v they_o to_o be_v equal_a in_o length_n though_o they_o be_v diverse_a in_o numbbe_n as_o for_o example_n in_o the_o last_o figure_n there_o be_v two_o parallel_n a._n d._n and_o e._n h_n and_o between_o they_o be_v draw_v three_o likeiamme_n the_o first_o be_v a._n b._n e._n f_o the_o second_o be_v e._n c._n d._n f_o and_o the_o third_o be_v c._n g._n h._n d._n the_o first_o and_o the_o second_o have_v one_o ground_n line_n that_o be_v e._n f._n and_o therefore_o in_o so_o much_o as_o they_o be_v between_o one_o pair_n of_o parallel_n they_o be_v equal_a 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